Carolyn and Paul are playing a game starting with a list of the integers $1$ to $n.$ The rules of the game are:

$\bullet$  Carolyn always has the first turn.

$\bullet$  Carolyn and Paul alternate turns.

$\bullet$  On each of her turns, Carolyn must remove one number from the list such that this number has at least one positive divisor other than itself remaining in the list.

$\bullet$  On each of his turns, Paul must remove from the list all of the positive divisors of the number that Carolyn has just removed.

$\bullet$  If Carolyn cannot remove any more numbers, then Paul removes the rest of the numbers.

For example, if $n=6,$ a possible sequence of moves is shown in this chart:

\begin{tabular}{|c|c|c|}
\hline
Player & Removed \# & \# remaining \\
\hline
Carolyn & 4 & 1, 2, 3, 5, 6 \\
\hline
Paul & 1, 2 & 3, 5, 6 \\
\hline
Carolyn & 6 & 3, 5 \\
\hline
Paul & 3 & 5 \\
\hline
Carolyn & None & 5 \\
\hline
Paul & 5 & None \\
\hline
\end{tabular}

Note that Carolyn can't remove $3$ or $5$ on her second turn, and can't remove any number on her third turn.

In this example, the sum of the numbers removed by Carolyn is $4+6=10$ and the sum of the numbers removed by Paul is $1+2+3+5=11.$

Suppose that $n=6$ and Carolyn removes the integer $2$ on her first turn. Determine the sum of the numbers that Carolyn removes.
The list starts as $1,$ $2,$ $3,$ $4,$ $5,$ $6.$

If Carolyn removes $2,$ then Paul removes the remaining positive divisor of $2$ (that is, $1$) to leave the list $3,$ $4,$ $5,$ $6.$ Carolyn must remove a number from this list that has at least one positive divisor other than itself remaining. The only such number is $6,$ so Carolyn removes $6$ and so Paul removes the remaining positive divisor of $6$ (that is, $3$), to leave the list $4,$ $5.$ Carolyn cannot remove either of the remaining numbers as neither has a positive divisor other than itself remaining.

Thus, Paul removes $4$ and $5.$

In summary, Carolyn removes $2$ and $6$ for a sum of $2+6=\boxed{8}$ and Paul removes $1,$ $3,$ $4,$ and $5$ for a sum of $1+3+4+5=13.$